$S_{12}$ and $P_{12}$-colorings of cubic graphs

If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partialG(x))=\partialH(y)$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. The Petersen coloring conjecture of Jaeger ($P{10}$-conjecture) states that for any bridgeless cubic graph $G$, one has: $P{10}\prec G$. The Sylvester coloring conjecture ($S{10}$-conjecture) states that for any cubic graph $G$, $S{10}\prec G$. In this paper, we introduce two new conjectures that are related to these conjectures. The first of them states that any cubic graph with a perfect matching admits an $S{12}$-coloring. The second one states that any cubic graph $G$ whose edge-set can be covered with four perfect matchings, admits a $P{12}$-coloring. We call these new conjectures $S{12}$-conjecture and $P{12}$-conjecture, respectively. Our first results justify the choice of graphs in $S{12}$-conjecture and $P{12}$-conjecture. Next, we characterize the edges of $P{12}$ that may be fictive in a $P{12}$-coloring of a cubic graph $G$. Finally, we relate the new conjectures to the already known conjectures by proving that $S{12}$-conjecture implies $S{10}$-conjecture, and $P{12}$-conjecture and $(5,2)$-Cycle cover conjecture together imply $P{10}$-conjecture. Our main tool for proving the latter statement is a new reformulation of $(5,2)$-Cycle cover conjecture, which states that the edge-set of any claw-free bridgeless cubic graph can be covered with four perfect matchings.